Decimals & Place Value

The photo finish

Two sprinters cross the line: one clocks $10.4$ seconds, the other $10.32$. Who won? The eye wants to say $10.32$ is bigger — it has more digits, and with whole numbers, longer always means larger. But $10.32$ is the winner here, a whole $0.08$ seconds ahead. Everything after a decimal point plays by rules the eye hasn’t fully learned yet, and this unit is about making those rules obvious instead of memorized.

Decimals just keep place value going

Start from what you’ve used since childhood: in $437$, each place is worth $10\times$ the place to its right — hundreds, tens, ones. Now just keep the pattern going. One step right of the ones must be worth ten times less: tenths ($\frac{1}{10}$). Another step: hundredths ($\frac{1}{100}$), then thousandths. The decimal point isn’t a wall — it’s just the marker for where “ones” ends. So $0.37$ means $3$ tenths $+\ 7$ hundredths, and in F3’s language that’s $\frac{37}{100}$: a decimal is a fraction whose denominator is a power of ten, with the denominator hidden in the position of the digits instead of written below them.

The widget opens on $42.857$ — before you look, say what the $8$ is worth, and what the $5$ is worth. Then type $0.307$: what is that middle $0$ doing? (It’s holding the tenths place open so the $3$ and $7$ don’t slide into the wrong units.)

Every decimal is a fraction in disguise

Decimal → fraction is just reading aloud: $0.6$ is “six tenths,” so $0.6 = \frac{6}{10} = \frac{3}{5}$. Fraction → decimal uses the other thing you know from F3 — a fraction is a division — so divide top by bottom: $\frac{3}{8} = 3 \div 8 = 0.375$.

But try $\frac{1}{3}$ and the division never ends: $0.333\ldots$, written $0.\overline{3}$ with a bar over the repeating block. Why do some fractions stop and others loop forever? The answer is the prime atoms from F2: a decimal place is a power of $10$, and $10$‘s atoms are exactly $2$ and $5$. If a fraction’s lowest-term denominator is built only from $2$s and $5$s — like $8 = 2^3$ — it can be renamed over a power of ten and the decimal terminates. Any other atom in the denominator ($3$, $7$, $11$…) can never divide a power of ten, so the division has no choice but to repeat.

Predict before converting: $\frac{7}{20}$ — terminate or repeat? ($20 = 2^2 \times 5$: only the right atoms.) Now $\frac{7}{12}$? ($12$ has a $3$ in it — expect a bar.)

Comparing: why “longer looks larger” fools you

Back to the sprinters. The instinct that $10.32 > 10.4$ comes from a lifetime of whole numbers, where an extra digit means another power of ten — $432$ really does beat $45$. But after the point, extra digits don’t add amount, they add fineness: $0.4$ and $0.40$ and $0.400$ are the same number sliced into thinner and thinner pieces. The fix is F3’s common-unit trick worn as a shortcut: pad with zeros until the lengths match — that’s renaming both numbers in the same unit — then compare tops. $10.40$ vs $10.32$: forty hundredths beats thirty-two.

Adding and subtracting: line up the points

Same principle, third appearance: only matching units can be counted together. Tenths add to tenths, hundredths to hundredths — so the decimal points must sit in one column. For $3.4 + 1.25$, pad to $3.40$, line up the points, and add columns: $4.65$. (Lining up the right-hand edges instead — the natural habit from whole-number addition — would add the $4$ tenths to the $5$ hundredths, which is unit nonsense.)

Multiplying: why you count decimal places

Here’s a place where decimal instinct underestimates strangeness: ask most people for $0.2 \times 0.3$ and "$0.6$" slips out, because $2 \times 3 = 6$ and the points feel decorative. Unmask the fractions and watch what really happens: $\frac{2}{10} \times \frac{3}{10} = \frac{6}{100} = 0.06$. The tenths multiply into hundredths — the denominators multiply too, which is exactly why the rule says: multiply as whole numbers, then give the answer as many decimal places as both factors had combined. And notice the answer is smaller than either factor: taking two-tenths of three-tenths shrinks it, the same “of” you met in F3.

Dividing: make the divisor whole

$1.5 \div 0.5$ asks “how many halves fit in one-and-a-half?” — three. The mechanical rule, shift both points right until the divisor is whole ($15 \div 5 = 3$), is legal for a reason you already own: shifting both is multiplying top and bottom by $10$, which is just making an equivalent fraction. The amount doesn’t change; only the costume does.

Run $0.2 \times 0.3$ and check the place-count in the steps. Then $1.5 \div 0.5$ — predict first: bigger or smaller than $1.5$? (Dividing by a number under $1$ grows the answer.)

Rounding: which marker is nearer?

Rounding just asks which tick of the ruler your number sits closest to. The familiar ”$5$ or more rounds up” rule is a shortcut for “is it at or past the halfway point?” — nothing more.

Round $42.857$ to the nearest tenth on the line — see which marker it hugs. Then try $0.95$ to the nearest tenth and watch the round-up ripple all the way into the ones place: $1.0$.

The one thing to remember

A decimal is a fraction over a power of ten with the denominator hidden in the digit positions — and every decimal rule is a fraction rule with the writing skipped. Comparing and adding need matching units (pad zeros, line up points); multiplying multiplies the hidden denominators too (count the places); dividing shifts both numbers into an equivalent, easier fraction.

What the digits mean

Every place is $10\times$ the one to its right. Left of the point: ones, tens, hundreds… Right of the point: tenths ($\frac{1}{10}$), hundredths ($\frac{1}{100}$), thousandths ($\frac{1}{1000}$)… So $0.37 = 3$ tenths $+\;7$ hundredths $= \frac{37}{100}$.

The rules

A rounding, step by step